Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning

TL;DR

This paper analyzes the approximation and statistical complexity of MultiNeural Operators in multi-task learning, showing they achieve near-optimal rates and are comparable to other architectures like DeepONet.

Contribution

It provides theoretical bounds and minimax rates for MultiNeural Operators, demonstrating their efficiency and equivalence to other multi-task operator learning methods.

Findings

Shared representations do not increase overall learning cost.

MultiNeural Operators achieve near-optimal approximation and generalization rates.

MNO and DeepONet have similar asymptotic approximation complexities.

Abstract

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.